Linear filters

In this figure the next definitions are used A - projection operator, B - pseudo-inverse operator for the image parameters a,( ), C - empirical posterior restoration of the FDD function w(a, ), E - optimal estimator. The projection operator A is non-observable due to the Kalman criteria [10] which is the main singularity for this problem. This leads to use the two step estimation procedure. First, the pseudo-inverse operator B has to be found among the regularization techniques in the class of linear filters. In the second step the optimal estimation d (n) for the pseudo-inverse image parameters d,(n) has to be done in the presence of transformed noise j(n). 

Karlstrdms, M., Magnusson Seger, M., Crack estimation with linear filtered tomosynthesis , ISSN 1400-3902, Dept of Electrical Engineering Linkoping, Sweden (1998). 

The solution is given by applying a linear filter f to the data and the Fourier transform of the solution writes ... 

To summarize, Wiener inverse-filter is the linear filter which insures that the result is as close as possible, on average and in the least squares sense, to the true object brightness distribution. 

Depending on how the previous measurements are combined in Eq. (7), univariate filtering methods can be categorized as linear or nonlinear. In terms of Eq. (7), linear filtering methods use a fixed scale parameter or are single-scale, whereas nonlinear filtering methods are multiscale. Figure 7 summarizes decompositions in terms of time and frequency. 

In single-scale filtering, basis functions are of a fixed resolution and all basis functions have the same localization in the time-frequency domain. For example, frequency domain filtering relies on basis functions localized in frequency but global in time, as shown in Fig. 7b. Other popular filters, such as those based on a windowed Fourier transform, mean filtering, and exponential smoothing, are localized in both time and frequency, but their resolution is fixed, as shown in Fig. 7c. Single-scale filters are linear because the measured data or basis function coefficients are transformed as their linear sum over a time horizon. A finite time horizon results infinite impulse response (FIR) and an infinite time horizon creates infinite impulse response (HR) filters. A linear filter can be represented as... 

Filter specifications are matched to the response times of the process. For example, a given process has a time constant of 5 minutes. That means that it can respond over the frequency range of 0 to 1/20 of a cycle per minute. Higher frequencies are attenuated naturally by the process. Thus, if the data contain components beyond 0.05 cycles per minute, then those components are likely to be unwanted interferences. The linear filter would pass the frequencies between 0 and 1/20 and reject frequencies outside this range. The filter should attenuate frequencies higher than one decade above the break-point frequency. Process measurements processed by this filter are transformed to a new sequence with less interference than the original data. In this way, an input mapping has been defined. 

R. E. Kalman, A new approach to linear filtering and prediction problems, Transactions of the ASME-Journal of Basic Engineering, Vol. 82, pp. 35-45,1960,... 

We might ask whether it is possible to create a linear filter function y(x) that could undo, by simple convolution, the smearing in the data caused by the spectrometer. We express the behavior of such a filter by the equation... 

This filter is not an inverse filter of the type that we seek, being intended only for noise reduction. It does not undo any spreading introduced by s(x). It is, however, an optimum filter in the sense that no better linear filter can be found for noise reduction alone, provided that we are restricted to the knowledge that the noise is additive and Gaussian distributed. 

Other modifications are possible to the same basic approach of seeking a filter that is optimum in the sense of least mean-square error. Backus and Gilbert (1970), for example, derive a linear filter by minimizing a sum of terms in which noise and resolution criteria are separately weighted. Frieden (1975) discusses variations of this technique. 

Nevertheless, certain types of prior knowledge can be introduced within the context of a linear method. Probabilities, signal and noise statistics, power spectra, and the like may be incorporated. Often this type of prior knowledge is difficult to obtain. In any case, it rarely exerts an influence nearly so profound as that of simple bounds on the amplitude of the solution. If the observing spread function obliterates all frequencies beyond the cutoff Q, they are forever lost to the linear restoration methods. No linear filter s... 

In this section we consider a model of interactions between the Kerr oscillators applied by J. Fiurasek et al. [139] and Perinova and Karska [140]. Each Kerr oscillator is externally pumped and damped. If the Kerr nonlinearity is turned off, the system is linear. This enables us to perform a simple comparison of the linear and nonlinear dynamics of the system, and we have found a specific nonlinear version of linear filtering. We study numerically the possibility of synchronization of chaotic signals generated by the Kerr oscillators by employing different feedback methods. 

Essentially, the ion storage trap is a spherical configuration of the linear quadrupole mass filter. The operations, however, differ in that the linear filter passes the sorted ions directly through to the detector, whereas the ion trap retains the unsorted ions temporarily within the trap. They are then released to the detector sequentially by scanning the electric field. These instruments are compact (benchtop), relatively inexpensive, convenient to use, and very sensitive. They also provide an inexpensive method to carry out GC/MS/MS experiments (Section 2.2.7) (GC is gas chromatography). 

The geometrical models allow the prediction of a room s early reverberant response, which will consist of a set of delayed and attenuated impulses. More accurate modeling of absorption and diffusion will tend to fill in the gaps with energy. Linear filters can be used to model absorption, and to a lesser extent diffusion, and allow reproduction of the directional properties of the early response. 

One frame-based implementation of this time-varying linear filtering is to filter windowed blocks of white noise and overlap and add the outputs over consecutive frames. A time-varying impulse response of a linear system can be associated with... 

In addition to obtaining correlograms, a large battery of methods are available to smooth time series, many based on so-called windows , whereby data are smoothed over a number of points in time. A simple method is to take the average reading over five points in time, but sometimes this could miss out important information about cyclicity especially for a process that is sampled slowly compared to the rate of oscillation. A number of linear filters have been developed which are apphcable to this time of data (Section 3.3), this procedure often being described as convolution. 

Smoothing functions can be introduced in a variety of ways, for example, as sums of coefficients or as a method for fitting local polynomials. In the signal analysis literature, primarily dominated by engineers, linear filters are often reported as a form... 

In Section 3.3 we discussed a number of linear filter functions that can be used to enhance the quality of spectra and chromatograms. When performing Fourier transforms, it is possible to apply filters to the raw (time domain) data prior to Fourier transformation, and this is a common method in spectroscopy to enhance resolution or signal to noise ratio, as an alternative to applying filters directly to the spectral data. 

---